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I have a question, reading a proof at some point there is written that the only inessential discriminant divisor of a cubic number field can only be $1$ and $2$, but i don't get why. If i'm not wrong the definition of inessential discriminant divisor is the follow: Given an algebraic number field $K$ of discriminant $ \delta_K$, denoting with $ \mathcal{O}_K$ his ring of integers, then one has for any $\alpha \in \mathcal{O}_K$ that $ \operatorname{disc}(\alpha)= [\mathcal{O}_K : \mathbb{Z}[\alpha] ]^2 \delta_K $, then it is natural to ask for which prime $p$ there exsits an element $\alpha $ such that $ p \nmid [\mathcal{O}_K : \mathbb{Z}[\alpha] ] $ in order to apply Kummer-Dedekind Theorem for find prime factorization of $ p \mathcal{O}_K $. So we say for a prime $p$ that is an inessential discriminant divisor if for any choice of $\alpha \in \mathcal{O}_K $ (but maybe for any $ \alpha \in K$, not sure) we have always that $ p \mid [\mathcal{O}_K : \mathbb{Z}[\alpha] ]$. Then a prime is not an inessential discriminant divisor if we can always find an element $ \alpha $ such that $ p \not\mid [ \mathcal{O}_K : \mathbb{Z}[\alpha]]$.

Now in the case that the field is monogenic then it is trivially true that for any prime we have that we can find always $ \alpha \in \mathcal{O}_K $ such that $ \mathcal{O}_K = \mathbb{Z}[\alpha] $, hence $ p \not\mid 1$. We may suppose then that the field is non monogenic, Dedekind gave an example for which a cubic number field is not monogenic and for which $2$ is a inessential discriminant divisor. But for $3 $ ? How to prove that $3$ cannot be a inessential discriminant divisor?

I proved that if $\alpha \in \mathcal{O}_K $ is an algebraic integer of degree $3$, and let $ P(x) $ be its minimal polynomial, if we suppose that $P$ is Eisenstein with respect to the prime $p$ then one has $ p \not\mid \left| \mathcal{O}_K/\mathbb{Z}[\alpha] \right| $, maybe can help.

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That $p \nmid [\mathcal O_K:\mathbf Z[\alpha]]$ when $\alpha$ has a minimal polynomial in $\mathbf Z[x]$ that is Eisenstein at $p$ doesn't help much, because such $p$ are a restricted type of ramified prime: $(p) = \mathfrak p^n$ where $n = [K:\mathbf Q]$ (a totally ramified prime in $K$). This does not address the more typical type of ramified prime in a cubic field where $(p) = \mathfrak p\mathfrak q^2$.

For a number field $K$, let $i(K)$ be the gcd of all the indices $[\mathcal O_K:\mathbf Z[\alpha]]$ for $\alpha \in \mathcal O_K$ such that $K = \mathbf Q(\alpha)$. It turns out that the prime factors of $i(K)$ are all small: they must be less than $[K:\mathbf Q]$: see Theorem 5.4 here.

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